Overscreened multichannel SU N Kondo model: Large-N solution and conformal field theory

Transcription

1 PHYSICAL REVIEW B VOLUME 58, UMBER 7 15 AUGUST 1998-I Overscreened multichannel SU Kondo model: Large- solution and conformal field theory Olivier Parcollet and Antoine Georges Laboratoire de Physique Théorique de l Ecole ormale Supérieure, 4 rue Lhomond, 7531 Paris Cedex 5, France Gabriel Kotliar and Anirvan Sengupta Serin Physics Laboratory, Rutgers University, Piscataway, ew Jersey 8854 Received ovember 1997 The multichannel Kondo model with SU() spin symmetry and SU(K) channel symmetry is considered. The impurity spin is chosen to transform as an antisymmetric representation of SU(), corresponding to a fixed number of Abrikosov fermions f f Q. For more than one channel (K1), and all values of and Q, the model displays non-fermi behavior associated with the overscreening of the impurity spin. Universal low-temperature thermodynamic and transport properties of this non-fermi-liquid state are computed using conformal field theory methods. A large- limit of the model is then considered, in which K/ and Q/q are held fixed. Spectral densities satisfy coupled integral equations in this limit, corresponding to a time-dependent saddle point. A low-frequency, low-temperature analysis of these equations reveals universal scaling properties in the variable /T, in agreement with conformal invariance. The universal scaling form is obtained analytically and used to compute the low-temperature universal properties of the model in the large- limit, such as the T residual entropy and residual resistivity, and the critical exponents associated with the specific heat and susceptibility. The connections with the noncrossing approximation and the previous work of Cox and Ruckenstein are discussed. S I. ITRODUCTIO AD MODEL Multichannel Kondo impurity models 1, have recently attracted considerable attention, for several reasons. First, in the overscreened case, they provide an explicit example of a non-fermi-liquid ground-state. Second, these models can be studied by a variety of controlled techniques, and provide invaluable testing grounds for theoretical methods dealing with correlated electron systems. One of the most recent and fruitful development in this respect has been the conformal field-theory approach developed by Affleck and Ludwig. 3 5 Finally, multichannel models have experimental relevance to tunneling phenomena in quantum dots and two level systems, 6 and possibly also to some heavy-fermion compounds. In this paper, we consider a generalization of the multichannel Kondo model, in which the spin symmetry group is extended from SU() to SU(). In addition, the model has a SU(K) symmetry among the K channels flavors of conduction electrons. We focus here on spin representations which are such that the model is in the non-fermi-liquid overscreened regime. We shall derive in this paper several universal properties of this SU() SU(K) Kondo model in the low-temperature regime. Specifically, we obtain the zerotemperature residual entropy, the zero-temperature impurity resistivity and T matrix, and the critical exponents governing the leading low-temperature behavior of the impurity specific heat, susceptibility, and resistivity. These results will be obtained using two different approaches. It is one of the main motivations of this paper to compare these two approaches in some detail. First Sec. III, we apply conformal field theory CFT methods to study the model for general values of and K. Then, we study the limit of large and K, with K/ fixed. This limit was previously considered by Cox and Ruckenstein, 7 in connection with the noncrossing approximation CA. There is a crucial difference between our work and that of Ref. 7, however, which is that we keep track of the quantum number specifying the spin representation of the impurity by imposing a constraint on the Abrikosov fermions representing the impurity which also scales proportionally to. As a result, the solution of the model at large follows from a true saddle-point principle, with controllable fluctuations in 1/. Hence, a detailed quantitative comparison of the large- limit to the CFT results can be made. The saddle-point equations are coupled integral equations similar in structure to those of the CA, except for the different handling of the constraint. The T impurity entropy and residual resistivity are obtained in analytical form in this paper from a lowenergy analysis of these coupled integral equations and shown to agree with the large- limit of the CFT results. We also demonstrate that the spectral functions resulting from these equations take a universal scaling form in the limit,t, which is precisely that expected from the conformal invariance of the problem. The Hamiltonian of the model considered in this paper reads H p K i1 1 J K A1 1 S A p c p i c p i p p i c p i t A c p. 1 i In this expression, c p i creates an electron in the conduction band, with momentum p, channel flavor index i 1,...,K, and SU() spin index 1,...,. The con /98/587/3794/$15. PRB The American Physical Society

2 PRB OVERSCREEED MULTICHAEL SU() KODO... duction electrons transform under the fundamental representation of the SU() group, with generators t (A A 1,..., 1). They interact with a localized spin degree of freedom placed at the origin S S A,A1,..., 1 which is assumed to transform under a given irreducible representation R of the SU() group. In the one-channel case (K1), and when R is taken to be the fundamental representation, this is the Coqblin- Schrieffer model of a conduction gas interacting with a localized atomic level with angular momentum j ( j1). 8 In this article, we are interested in the possible non-fermi-liquid behavior associated with the multichannel generalization (K1). 1, We shall mostly focus on the case where R corresponds to antisymmetric tensors of Q indices, i.e., the Young tableau associated with R is made of a single column of Q indices. In that case, it is convenient to use an explicit representation of the localized spin in terms of species of auxiliary fermions f (1,...,), constrained to obey 1 f f Q so that the 1 traceless components of S can be represented as S f f (Q/). For these choices of R, the Hamiltonian can be written as after a reshuffling of indices using a Fierz identity H p K i1 1 p c p i c p i J K f f Q p p c p i c p. 3 i i In a recent paper, 9 the case of a symmetric representation of the impurity spin corresponding to a Young tableau made of a single line of P boxes has been considered by two of us. In that case, a transition from overscreening to underscreening is found as a function of the size P of the impurity spin. In contrast, the antisymmetric representations considered in the present paper always lead to overscreening except for K1 which is exactly screened, as shown below. As long as only the overscreened regime is considered, the analysis of the present paper applies to symmetric representations as well, up to some straightforward replacements. II. STROG-COUPLIG AALYSIS It is easily checked that a weak antiferromagnetic coupling (J K ) grows under renormalization for all K and, and all representations R of the local spin. What is needed is a physical argument in order to determine whether the renormalization group RG flow takes J K all the way to strong coupling underscreened or exactly screened cases, or whether an intermediate non-fermi-liquid fixed point exists overscreened cases. Following the ozieres and Blandin 1 analysis of the multichannel SU() model, we consider the strong-coupling fixed point J K. In this limit, the impurity spin binds a certain number of conduction electrons at most K because of the Pauli principle. The resulting bound-state corresponds FIG. 1. Young tableau corresponding to the strong-coupling state. to a new spin representation R sc which is dictated by the minimization of the Kondo energy. For the specific choices of R above, we have proven the following. i In the one-channel case (K1) and for arbitrary and Q, R sc is the singlet representation of dimension d(r sc )1. It is obtained by binding Q conduction electrons to the Q pseudofermions f. The impurity spin is thus exactly screened at strong-coupling. ii For all multichannel cases K, arbitrary and Q), the ground state at the strong-coupling fixed point is the representation R sc characterized by a rectangular Young tableau with Q lines and K1 columns. Its dimension d(r sc ) i.e, the degeneracy of the strong-coupling bound state is larger than the degeneracy at zero coupling, given by d(r)( Q )!/Q!(Q)!. The Young tableau associated with the strong-coupling state in both cases is depicted in Fig. 1. The detailed proof of these statements and the explicit construction of R sc are given in Appendix A. These properties are sufficient to establish the following. i In the one-channel case, the strong-coupling fixed point is stable under RG, and hence the impurity spin is exactly screened by the Kondo effect. ii In the multichannel case, a direct RG flow from weak to strong coupling is impossible, thereby suggesting the existence of an intermediate coupling fixed point overscreening. The connection between these statements and the above results on the nature and degeneracy of the strong-coupling bound state is clear on physical grounds. Indeed, it is not possible to flow under renormalization from a fixed point with a lower ground-state degeneracy to a fixed point with a higher one, because the effective number of degrees of freedom can only decrease under RG. Hence, no flow away from the strong-coupling fixed point is possible in the one-channel (K1) case since the strong-coupling state is nondegenerate. Also, no direct flow from weak to strong coupling is possible for K since d(r sc )d(r). These statements can be made more rigorous 5 by considering the impurity entropy defined as S imp lim lim STS bulk T, T V 4

3 3796 PARCOLLET, GEORGES, KOTLIAR, AD SEGUPTA PRB 58 where S bulk denotes the contribution to the entropy which is proportional to the volume V and is simply the contribution of the conduction electron gas, and care has been taken in specifying the order of the infinite-volume and zerotemperature limits. At the weak-coupling fixed point S imp (J K )ln d(r), while S imp (J K )ln d(r sc ) at strong-coupling. S imp must decrease under renormalization, 5 a property which is the analog for boundary critical phenomena to Zamolodchikov s c theorem in the bulk. This suggests a RG flow of the kind indicated above. This conclusion can of course be confirmed by a perturbative calculation in the hopping amplitude around the strong-coupling fixed point. 1 The value of S imp will be calculated below at the intermediate fixed point in the overscreened case, and found to be noninteger as in the case 5. III. COFORMAL FIELD THEORY APPROACH Having established the existence of an intermediate fixed point for K, we sketch some of its properties that can be obtained from conformal field-theory CFT methods. This is a straightforward extension to the SU() case of Affleck and Ludwig s approach for SU(). 3,4 The aim of this section is not to present a complete conformal field-theory solution, but simply to derive those properties which will be compared with the large- explicit solution given below. In the CFT approach, the model 1 is first mapped at low energy onto a (11)-dimensional model of K chiral fermions. At a fixed point, this model has a local conformal symmetry based on the Kac-Moody algebra SÛ K () s SÛ (K) f Û(1) c corresponding to the spin-flavor-charge decomposition of the degrees of freedom. The free-fermion spectrum at the weak-coupling fixed point can be organized in multiplets of this symmetry algebra: to each level corresponds a primary operator in the spin, flavor, and charge sectors. A major insight 3 is then that the spectrum at the infrared stable, intermediate coupling fixed point can be obtained from a fusion principle. Specifically, the spectrum is obtained by acting, in the spin sector, on the primary operator associated with a given free-fermion state, with the primary operator of the SÛ K () s algebra corresponding to the representation R of the impurity spin leaving unchanged the flavor and charge sectors. The fusion rules of the algebra determine the new operators associated with each energy level at the intermediate fixed point. This fusion principle also relates the impurity entropy S imp as defined above to the modular S matrix S R of the SÛ K () algebra this is the matrix which specifies the action of a modular transformation on the irreducible characters of the algebra corresponding to a given irreducible representation R). Specifically, denoting by R the trivial identity representation S imp ln S R S. The expression of the modular S-matrix for SÛ K () can be found in the literature. 1 We have found particularly useful to make use of an elegant formulation introduced by Douglas, 11 which is briefly explained in Appendixes A and B. For the 5 representations R associated to a single column of length Q corresponding to Eq., one finds using this representation Q sin1n/k S imp ln n1. 6 sinn/k It is easily checked that indeed S imp ln d(r) for all values of, Q, and K. ote also that this expression correctly yields S imp in the exactly screened case K1 for arbitrary,q). The low-temperature behavior of various physical quantities can also be obtained from the CFT approach. At the intermediate coupling fixed point, the local impurity spin acquires the scaling dimension of the primary operator of the SÛ K () s algebra associated with the ( 1)-dimensional adjoint representation of SU(). Its conformal dimension s such that S()S(t)1/t s at T reads s K. Integrating this correlation function, this implies that the local susceptibility loc 1/T S()S()d corresponding to the coupling of an external field to the impurity spin only diverges at low temperature when K, while it remains finite for K: K: loc T 1 K/K, K: loc ln 1/T, K: loc const. Exactly at the fixed point, the singular contributions to the specific heat and impurity susceptibility imp bulk defined by coupling a magnetic field to the total spin density vanish. 3,1 Indeed, the singular behavior is controlled by the leading irrelevant operator compatible with all symmetries that can be generated. An obvious candidate for this operator is the spin, flavor, and charge singlet obtained by contracting the spin current with the adjoint primary operator above, J 1. It has dimension 1 /(K) 1 s. This leads to a singular contribution to the impurity susceptibility arising from perturbation theory at second order in the irrelevant operator 3,1 of the same nature than for loc given in Eq. 8: imp loc. Another irrelevant operator can be constructed in the flavor sector in an analogous manner, namely, J 1 f. This f operator has dimension K/(K), which is thus lower than the above operator in the spin sector when K. This operator could a priori contribute to the low-temperature behavior of the specific heat, which would lead to a divergent specific heat coefficient C/T for both K and K. On the basis of the explicit large- calculation presented below, we believe, however, that this operator is not generated for model 1 when the conduction band density of state DOS is taken to be perfectly flat and the cutoff is taken to infinity conformal limit, so that the specific heat ratio has the same behavior as the susceptibilities above: C/T loc imp.if 7 8

4 PRB OVERSCREEED MULTICHAEL SU() KODO... the model is extended to an impurity spin with internal flavor degrees of freedom, this operator will however show up leading to C/TT (K)/(K) for K. It also appears see Sec. VI B if an Anderson model generalization of model 1 is considered away from particle-hole symmetry. IV. SADDLE-POIT EQUATIOS I THE LARGE- LIMIT We now turn to the analysis of the large- limit of this model. This will be done by setting The quartic term in this expression can be decoupled formally using two bilocal fields Q(,) and Q (,) conjugate to i B i ()B i () and f () f (), respectively, leading to the action S d f f 1 J d i B i B i d i f f q K, J K J 9 ddq,g 1 Q, and taking the limit for fixed values of and J, so that the number of channels is also taken to be large. In Ref. 7 see also Ref., Cox and Ruckenstein considered this limit while holding Q fixed (Q1). They obtained in this limit identical results to those of the noncrossing approximation CA. 13 Here, we shall proceed in a different manner by taking Q to be large as well: Q q. 1 This ensures that a true saddle point exists, with controllable fluctuations order by order in 1/. It will also allow us to study the dependence on the representation R of the local spin, parametrized by q. 14 The approach of Ref. 7 is recovered in the limit q or 1). The action corresponding to the functional integral formulation of model 3 reads S d d i d c i G 1 c i f f d i f f q J d c i c i i f f q. 11 In this expression, the conduction electrons have been integrated out in the bulk, keeping only degrees of freedom at the impurity site. G (i n ) p 1/(i n p ) is the on-site Green s function associated with the conduction electron bath. In order to decouple the Kondo interaction, an auxiliary bosonic field B i () is introduced in each channel, and the conduction electrons can be integrated out, leaving us with the effective action S eff d f f d i 1 J d i f f q B i B i 1 d d dd Q, i B i B i dd Q, f f. 13 The B and f fields can now be integrated out to yield SddQ,G 1 Q, q d i Tr ln i Q,K Tr ln 1 J Q,. 14 This final form of the action involves only the three fields Q, Q, and, and scales globally as thanks to the scalings K and Qq. Hence, it can be solved by the saddlepoint method in the large- limit. At the saddle point, sp ()i becomes static and purely imaginary, while Q sp Q() and Q sp Q () retain time dependence but depend only on the time difference they identify with the bosonic and fermionic self-energies, respectively. The final form of the coupled saddle-point equations for the fermionic and bosonic Green s functions G f () Tf()f (), G B ()TB()B () and for the Lagrange multiplier field read: f G G B, B G G f, 15 where the self-energies f and B are defined by G 1 f i n i n f i n, G 1 B i n 1 J Bi n. 16 In these expressions n (n1)/ and n n/ denote fermionic and bosonic Matsubara frequencies. Let us note that the field B() is simply a commuting auxiliary field rather than a true boson its equal-time commutator vanishes. As a result G B () isa-periodic function, but does not share the usual other properties of a bosonic Green function in particular at high frequency. Finally, is determined by the third saddle-point equation i B i f G B i f. 1 G f 1 n G f i n e i n q. 17

5 3798 PARCOLLET, GEORGES, KOTLIAR, AD SEGUPTA PRB 58 These equations are identical in structure to the usual CA equations, except for the last equation 17 which implements the constraint and allows us to keep track of the choice of representation for the impurity spin. V. SCALIG AALYSIS AT LOW FREQUECY AD TEMPERATURE A. General considerations The analysis of the CA equations in Ref. 15 can be applied in order to find the behavior of the Green s functions 1 in the low temperature, long time regime defined by T K where T K is the Kondo temperature. In this regime, a power-law decay of the Green s functions is found: G f () A f, G B A B, T 1 f K. B 18 The scaling dimensions f and B can be determined explicitly by inserting this form into the above saddle-point equations and making a low-frequency analysis, as explained in Appendix C. This yields f 1 1, B The overall consistency of Eqs. 15, 16 at large time also constrains the product of amplitudes A f A B Eq. C8 in Appendix C and dictates the behavior of the self-energies denoting by ImG (i )/ the conduction bath density of states at the Fermi level f A B 1 B 1, B A f 1 f 1 together with the sum rule B, 1 J. 1 The expression 19 of the scaling dimensions f and B is in complete agreement with the CFT result. Indeed, the fermionic field transforms as the fundamental representation of the SÛ() K spin algebra, while the auxiliary bosonic field transforms as the fundamental representation of the SÛ(K) flavor algebra, leading to f ( 1)/(K) and B (K 1)/K(K), which agree with Eq. 19 in the large- limit. Also, the local impurity spin correlation function, given in the large- limit by S()S() G f ()G f () is found to decay as 1/ s, with s f 1/(1) in agreement with the CFT result 7. As will be shown below, however, these asymptotic behaviors at T do not provide enough information to allow for the computation of the low-temperature behavior of the impurity free-energy and to determine the T impurity entropy 4. One actually needs to determine the Green s functions in the limit,, but for an arbitrary value of the ratio / i.e., to analyze the low-temperature, lowfrequency behavior of Eqs. 15, 16 keeping the ratio /T fixed 16. It is easily seen that in this limit the Green s functions and their associated spectral densities f,b () (1/)ImG f,b (i ) obey a scaling behavior for,t 1 K, with / arbitrary G f,b A f,b f,bg f,b a f A f T f 1 f T, B A B T B 1 B T. b In these expressions, g f,b and f,b are universal scaling functions which depend only on and q and not on the specific shape of the conduction band or the cutoff. These scaling functions will now be found in explicit form. B. The particle-hole symmetric representation q 1/ We shall first discuss the case where the representation R has Q/ boxes (q 1/), for which there is a particlehole symmetry among pseudofermions under f f. The expression of the scaling functions g f,b in that case can be easily guessed from general principles of conformal invariance. The idea is that, in the limit T 1 K with / fixed, the finite-temperature Green s function can be obtained from the T Green s function by applying the conformal transformation zexp(i /). 17 Applying this to the T power-law decay given by Eq. 18, one obtains the wellknown result for the scaling functions ( /): g f ;q 1/ sin f, g B ;q 1/ sin B 3 with the periodicity requirements g f ( 1)g f ( ),g B ( 1)g B ( ). ote that these functions satisfy the additional symmetry g f,b (1 )g f,b ( ) indicating that they can only apply to the particle-hole symmetric case q 1/. The corresponding form of the scaling functions associated with the spectral densities b reads, with /T after a calculation detailed in Appendix C f,q 1/ 1 f 1 cosh fi / f i /, f 4 B (,q 1/) 1 () B 1 sinh [ Bi ( /)][ B i ( /)]. ( B )

6 PRB OVERSCREEED MULTICHAEL SU() KODO... ote that the singularity of the T case is now recovered for T: f,b,q 1/ 1 f,b 1 1 f,b. 5 It is an interesting calculation, performed in detail in Appendix C, to check that indeed these scaling functions do solve the large- equations 15,16 at finite temperature in the scaling regime. That CA-like integral equations obey the finite-temperature scaling properties dictated by conformal invariance has not, to our knowledge, been pointed out in the previous literature. C. General values of q 1. Spectral asymmetry Let us move to the general case of representations with q 1 in which the particle-hole symmetry between pseudofermions is broken. The exponent of the power-law singularity in the T spectral densities is not affected by this asymmetry. It does induce, however, an asymmetry of the prefactors associated with positive and negative frequencies as. We introduce an angle to parametrize this asymmetry, defined such that f h, sin f, 1 f f h, sin f, 6 1 f where h(,) is a constant prefactor. The explicit dependence of on q will be derived below. This corresponds to the following analytic behavior of the Green s function in the complex frequency plane, as z : G R f zh, ei f i Im z. z 1 f 7 Equivalently, this means that the symmetry G f () G f () is broken, and that the scaling function g f ( ) must satisfy from the behavior of its Fourier transform g f g f 1 sin f sin f. 8 We have found, by an explicit analysis of the saddle-point and constraint equations in the scaling regime, which is detailed in Appendix C, that the full scaling functions for this asymmetric case are very simply related to the symmetric ones at q 1/, through g f,b ;q e 1/ cosh/ g f,b ;q 1/ e 1/ f,b, 9 cosh/ sin FIG.. Plot of f and b as a function of for different values of the asymmetry parameter :,, 5, ( f.3). where the parameter is simply related to so as to obey Eq. 8: ln sin/1 sin/1. 3 Fourier transforming, this leads to the scaling functions for the spectral densities cosh / f,q cosh /cosh/ f,q 1/, sinh / B,q sinh /cosh/ B,q 1/. 31 The thermal scaling function for the fermionic spectral density f and the bosonic one b are plotted in Fig. for various values of the asymmetry parameter. We also note the expression for the maximally asymmetric case corresponding, as shown below, to q, i.e., to the limit Q as in the usual CA:

7 38 PARCOLLET, GEORGES, KOTLIAR, AD SEGUPTA PRB 58 f e / f1 f i / f i /. 3 f We also note, for further use, the expressions of the full Green s functions in the complex frequency plane defined by g f /B (z,) d f /B ( )/(z ), in the scaling regime for Im z: i f 1 g f z, cosh/ f sin f f i z f i z cos f i sin f i z, 33 B 1 g B z, cosh/ B sin B B i z B i z sin B i sin B i z. 34 The reader interested in the details of these calculations is directed to Appendix C. At this stage, the point which remains to be clarified is the explicit relation between the asymmetry parameter and the parameter q specifying the representation. This is the subject of the next section. Before turning to this point, we briefly comment on the CFT interpretation of the asymmetry parameter or ) associated with the particle-hole asymmetry of the fermionic fields. The form 3 of the correlation functions at finite temperature in the scaling limit can be viewed as those of the exponential of a compact bosonic field with periodic boundary conditions. The asymmetric generalization 9 corresponds to a shifted boundary condition on the boson i.e., to a twisted boundary condition for its exponential.. Relation between q and Let us clarify the relation between the spectral asymmetry parameter, and the parameter q specifying the spin representation. That such a relation exists in universal form is a remarkable fact: indeed is a low-energy parameter associated with the low-frequency behavior of the spectral density, while q is the total pseudofermion number related by the constraint equation 17 to an integral of the spectral density over all frequencies. The situation is similar to that of the Friedel sum rule in impurity models, or to Luttinger theorem in a Fermi liquid, and indeed the derivation of the relation between q and follows similar lines. 18 It is in a sense a Friedel sum rule for the quasiparticles carrying the spin degrees of freedom namely, the pseudofermions f. We start from the constraint equation 17 written at zerotemperature as d q i lim t G fe it. 35 In this expression, and below in this section, G f () and G B () denote the Feynman T Green s functions while the retarded Green s functions are denoted by G R. Using analytic continuation of Eq. 16, we have ln G f G f f G f, 36 ln G B G B B, 37 so that Eq. 35 can be rewritten as q i lim d t ln G f G f f ln G B G B B eit. 38 In this expression, the bosonic part which vanishes altogether has been included in order to transform further the terms involving derivatives of the self-energy, using analyticity. This transformation is only possible if both fermionic and bosonic terms are considered. This is because the Luttinger-Ward functional 18 of this model involves both Green s functions. It has a simple explicit expression which reads dtg tg f, tg B,i t LW G f,,g B,i,i 39 such that the saddle-point equations 15 are recovered by derivation f, t LW G f, t, B,it LW G B,i t. 4 From the existence of LW, we obtain the sum rule d f G f B G B. 41 ote that there is no logarithmic divergence at in this expression. After integrating by parts and using the asymptotic behavior of the two Green s functions in order to eliminate the boundary terms, we get d q i ln G f ln G B e i. 4 Since G()G R () for and G()G R () for with G R the retarded Green s function, this can be transformed using

8 PRB OVERSCREEED MULTICHAEL SU() KODO... d ln G f,b d ln G e R i f,b e i d ln G R f,b e G R f,b i. 43 The first integrals in the right hand side can be deformed in the upper plane and their sum vanishes. 18 Thus we obtain denoting by arg G the argument of G) q arg G f R arg G f R arg G B R arg G B R. 44 arg G f R ( ) directly follows from the parametrization 7 defining. It can also be read off from the behavior of the scaling function g f (z) for z. Thus, from Eqs we can also read off arg G B R ( ): arg G R f f, arg G R B f. 45 Taking into account that G R f () 1/ and that Im G R f we have arg G R f (). Similarly, we have arg G R B (). Inserting these expressions into Eq. 44, we finally obtain the desired relation between q and or ): 1 1 q, sinq /1 ln sin1q /1. 46 This, together with Eq. 33, fully determines the universal scaling form of the spectral functions in the low-frequency, low-temperature limit. VI. PHYSICAL QUATITIES AD COMPARISO WITH THE CFT APPROACH A. Impurity residual entropy at T The impurity contribution to the free-energy per color of spin f imp (FF bulk )/ reads, at the saddle point f imp q T n ln G f i n T n ln G B i n d f G f. 47 This expression can be derived either directly from the saddle-point effective action in which case the last term arises from the quadratic term in Q and Q, or from the relation between the free-energy and the Luttinger-Ward functional. F bulk T Tr ln G is the free energy of the conduction electrons. In Eq. 47 the formulas Tr ln G are ambiguous. We must precisely define which regularization of these sums we consider: the actual value of the sums depends on the precise definition of the functional integral. For the fermionic field, the standard procedure of adding and substracting the contribution of a free local fermion, and introducing an oscillating term to regularize the Matsubara sum holds: Tr ln G f T ln T n lni n G f i n e i n. 48 The situation is somewhat less familiar for the bosonic field. As pointed out above, the latter is merely a commuting auxiliary field rather than a true boson. We have found that the correct regularization to be used is Tr ln G B T lim n n lnjg B i n. 49 The factor of J takes into account the determinant introduced by the decoupling with B, and a symmetric definition of the convergent Matsubara sum has been used. Some details and justifications about these regularizations are given in Appendix D. We shall perform a low-temperature expansion of Eq. 47, considering successively the particle-hole symmetric (q 1/) and asymmetric (q 1/) cases, which require rather different treatments. 1. The particle-hole symmetric point q 1 In this case, so that the first term in Eq. 47 does not contribute. Let us consider the last term in 47. Using the spectral representation of G f and the definition of f we obtain easily for df G f We substract the value at T: f d 1e. TT df,t f,t d f 1e In the second term, we can replace f by its scaling limit. So this term is of order O(T f 1 ). We know the asymptotics of f : f (x) x C 1 x f 1 C x f 3. The term x f cancels due to the particle-hole symmetry. Thus, the first term in Eq. 51 is of the form T f 1 dxx f (x) C 1 x f 1 the integral is convergent. We conclude that (T)()O(T f 1 ), so that the last term in Eq. 47 does not contribute to the zero-temperature entropy in the particle-hole symmetric case. Let us express the remaining terms in Eq. 47 as integrals over real frequencies, using the regularizations introduced above. As detailed in Appendix D, this leads to the following expression, involving the argument of the finitetemperature retarded Green s functions:

9 38 PARCOLLET, GEORGES, KOTLIAR, AD SEGUPTA PRB 58 f imp 1 d F arctan n G f G f n B arctan G B G B. 5 In these expressions n F respectively, n B are the Fermi respectively, Bose factor. At this point, it would seem that in order to perform a low-temperature expansion of the free-energy, one has to make a Sommerfeld expansion of the Fermi and Bose factors. This is not the case, however, for two reasons: i the argument of the Green s functions appearing in Eq. 5 are not continuous at, so that a linear term in T does appear as expected from the nonzero value of S imp and ii the Green s functions have an intrinsic temperature dependence, and the full scaling functions computed above must be used in Eq. 5. More precisely, when computing the difference f imp (T) f imp (T), the leading term is obtained by replacing the Green s function by their scaling form 33. These considerations lead to the following expression of the impurity entropy per spin color s imp S imp / at zero temperature for q 1 : s imp 1 1 d af af 1 d e 1 a f sgn d ab ab s imp q 1 ln 1 tan/1 ln1u 1u du. 57 This can also be rewritten, after a change of integration variable, as s imp q 1/ 1 S imp 1 f 1 f 1 with f x x ln sinudu. 58 This coincides with the large- limit of the CFT result, Eq. 6, in the particle-hole symmetric case.. The general case q 1/ For q 1, the first term in Eq. 47 also contributes to the entropy. Indeed, as shown below, the Lagrange multiplier (T) at the saddle point has a term which is linear in temperature. This stems from a very general thermodynamic relation, which is derived by taking the derivative of Z with respect to q in the functional integral, leading to F 1 i, q 59 where the average is to be understood with the action 11. At the saddle point, we thus have f imp /q and in particular 1 d a b sgn e TT s imp q. 6 In this expression, a f,b denotes the arguments of the scaling functions, obtained from Eq. 33: a f arctan g f g f arctan cot f tanh, a B arctan g B g B arctan tan f tanh. From Eq. 53, we obtain with ttan f s imp 1 1 du arctant 1u uarctan u t arctanut u arctanut u1u To perform the integration, we note that /t (/t)(1 t )s imp /(1) t/(1t ) and we obtain finally the simple expression We shall directly use this equation in order to compute the residual entropy, by calculating the linear correction in T to, and then integrating over q. This method shortcuts the full low-temperature expansion of the free energy as done in the previous section, which actually turns out to be quite a difficult task to perform correctly for q 1/. 19 In order to calculate this linear correction, we shall relate to the highfrequency behavior of the fermion Green s function. As f (i n ) when n we have G f i n 1 i n i n O 1 i n. 61 This shows that is the discontinuity of the derivative of G f () with respect to at : G f G f Let us define g() by G f e/1/ cosh / g, df. 6 63

10 PRB OVERSCREEED MULTICHAEL SU() KODO... where is the spectral asymmetry parameter in Eq. 46 at this stage we emphasize that the full finite temperature, finite cutoff, Green s function G f is considered. Equation 6 can be rewritten as Ttanh g g g g, 64 where we have used that G f ( )G f ( )1. Denoting by g () the spectral function associated with g, we have g g d g g 1e. 65 In the scaling limit, the spectral function g must become particle-hole symmetric since the effect of the particle-hole asymmetry in this limit is entirely captured by in Eq. 63, and must coincide with A f T f 1 f (/T;q 1/). Hence, following the same reasoning than for above, the term in Eq. 65 is of order consto(t f 1 ). Thus we have TT A TT, 66 where A g( ) g( ) is the discontinuity of the derivative g. A reflects the particle-hole asymmetry of g and thus vanishes in the scaling limit. Actually the derivative A/T also vanishes as T as we now show. Consider first sending the bare cutoff to infinity along with J) soasto keep the Kondo temperature fixed. In this limit A takes the form: ATf(T/T K ). The low-energy scaling limit, in which Eq. 9 holds, can be reached by fixing T and sending T K to infinity. Since g must become particle-hole symmetric in this limit, this implies that f (x) vanishes at small x. Hence, taking a derivative with respect to temperature, of A Tf(T/T K ) we find that A/T T. Thus we finally obtain s imp, 67 q TT where (q ) is given in Eq. 46. Integrating this equation over q taking into account as a boundary condition the value of s imp (q 1/) obtained above, we finally derive the expression of the entropy s imp 1 S imp 1 f 1 f 1 1q f 1 q. 68 with, as above, f (x) x ln sin(u)du. The expression 68 coincides precisely with the large limit of the CFT result 6. A plot of the residual entropy and of the asymmetry parameter as a function of q is displayed in Fig. 3. s imp is maximal at q 1/ and vanishes as q as expected. In this section, we have discovered that the spectral asymmetry parameter twist shares a fairly simple relation with the residual entropy, given by Eq. 67. These are two FIG. 3. Residual entropy s imp and vs q for 1.5. universal quantities, characteristic of the fixed point. Remarkably, also coincides with the term proportional to T in while itself is nonuniversal, its linear term in T is. Itis tempting to speculate that a deeper interpretation of these facts is still to be found. B. Internal energy and specific heat The low-temperature behavior of the internal energy in the large- limit can be obtained by two different methods. We shall briefly describe both since they emphasize different and complementary aspects of the physics. In the first method, we use the effective action in the form 11, before the decoupling with the auxiliary bosonic field B i () is made. We thus have a quartic interaction vertex between the conduction electrons at the origin and the Abrikosov fermions representing the quasiparticles in the spin sector, which reads J f f Q K 1, c i c i. 69 i1 One can then perform a skeleton expansion of the freeenergy functional in terms of the interacting Green s functions for the pseudofermions and the conduction electrons G f () and G c (). The first-order Hartree contribution to this functional vanishes because the spin operator in Eq. 69 is written in a traceless manner. The next contribution, at second order, yields the most singular contribution at low temperature and reads EJ dgc G f. 7 At the saddle point, the interacting conduction electron Green s function is G c ()G f ()G B (), and hence its dominant long-time behavior is G c ()1/. Inserting this, together with G f ()1/ f in Eq. 7, we see that the leading low-temperature behavior to the energy reads E c 1 T 4 f 1 c T, and hence to the specific heat coefficient

11 384 PARCOLLET, GEORGES, KOTLIAR, AD SEGUPTA PRB 58 1: C/TT 4 f 1 1: C/Tln 1/T, 1: C/T const T 1 1/1, 71 which agrees with the CFT result described above. We note that there is a quite precise connection between this calculation and the CFT approach: the operator appearing in the Kondo interaction 69 acquires conformal dimension ( c f )1 f and has the appropriate structure of the scalar product of a spin current with the operator S transforming as the adjoint. Therefore, it is the large- version of the leading irrelevant operator associated with the spin sector, as described in Sec. III. It is satisfying that the leading low-t behavior comes from the second-order contribution of this operator in this formalism as well. We note that for the simple Kondo model 1, in the scaling limit, the analogous irrelevant operator in the flavor sector does not show up in the calculation of the energy in the large- solution. We shall comment further on this point below. The second method to investigate the internal energy is to push the low-temperature expansion of the free energy to higher orders. To this end, we need to compute higher-order terms in the expansion a of the Green s functions in the scaling regime. This computation is detailed in Appendix C 3, and leads to G f A f fg f 4 fg f, We also note that this behavior of the specific heat is modified when an Anderson model version of the present model is considered as in Ref. 7. Because the noninteracting slave boson propagator has a frequency dependence, the exponents of the second-order terms as written in Eq. 7 are only correct for 1 for the Anderson model. For 1, the term 4 fg f () (/) is replaced by 1 g f () (/), while 1 g B () (/) is replaced by 4 bg B () (/). As a result, one finds a diverging specific heat coefficient in both cases, with C/TT (1)/(1) for 1 and C/TT (1)/(1) for 1. The behavior for 1 is due to the leading irrelevant operator in the flavor sector. Similarly, for 1 in the Anderson model, the susceptibility associated with the flavor (channel) sector f is found to diverge, 7 so that a finite Wilson ratio can still be defined as T f /C for 1. C. Resistivity and T matrix In order to discuss transport properties, we define a scattering T matrix for the conduction electrons in the usual manner for a single impurity: Gk,k,i G k,i k,k G k,i TG k,i, 74 where G and G denote the interacting and noninteracting conduction-electron Green s functions, respectively. Taking a flat particle-hole symmetric band for the conduction electron and denoting by the local noninteracting density of states, this yields the local conduction electron Green s function in the form 6 fg 3 f G B A B Bg B 1 g B 1 fg B Let us emphasize that the exponents appearing in this expansion are not symmetric between the bosonic and fermionic degrees of freedom. This is because we are dealing with the Kondo model for which the auxiliary field bosonic propagator has no frequency dependence in the noninteracting theory. Also, the expansion given in Eq. 7 assumes a perfectly flat conduction band in the limit of an infinite bandwith conformal limit. Using this expansion into the expression 5 of the free energy leads to a specific heat coefficient C/Tc T f 1 c 1 T 4 f 1 c. The coefficient c actually vanishes, so that the behavior in Eq. 71 is recovered. The vanishing of c was clear in the first approach, where it followed from the absence of Hartree terms. In the CFT approach, it is associated with the fact that the leading irrelevant operator does not contribute to the free energy at first order. The vanishing of c implies nontrivial sum rules relating the scaling functions g f,b and g () f,b which we have not attempted to check explicitly. Gi Gk,ki 1i T. k,k 75 Following Ref. 4, we parametrize the zero-frequency limit of the T matrix in terms of a scattering amplitude S 1 as T i 1S 1 so that, the zero frequency electron Green s function reads 76 1S 1 Gi i. 77 S 1 1 corresponds to no scattering at all, while S 1 1 corresponds to maximal unitary scattering i.e., / phase shift and vanishing conduction electron density of states at the impurity site. In the overscreened case, as noted in Ref. 4, S 1 is in general such that S 1 1, reflecting the non-fermiliquid nature of the model, and the fact that the actual quasiparticles bear no resemblance to the original electrons. In addition here, we shall find the feature that S 1 is in fact a complex number for nonparticle-hole symmetric spin representations i.e., q 1/.

12 PRB OVERSCREEED MULTICHAEL SU() KODO... We first derive an expression for S 1 for arbitrary, K and spin representation Qq by generalizing to SU() the CFT approach of Ref. 4. There, it was shown that S 1 can be expressed as a ratio of elements of the modular S matrix S, of the SÛ() K algebra. Denoting by the identity representation, by F the fundamental representation corresponding to a Young tableau with a single box, and by R the representation in which the impurity lives Young tableau with a single column of Q boxes, one has 3,4 S 1 S F,R /S,R. 78 S F, /S, The evaluation of these elements of the modular S matrix can be done along the same lines as the conformal field theory calculation of the entropy, described above. Some details are given in Appendix B. The result is: S 1 sin1/kexpi 1q /Ksin/Kexpi 11q /K. sin/k 79 otice that S 1 has both real and imaginary parts in the absence of particle hole symmetry q 1. Let us take the large- limit of this expression, with K/ fixed. This reads, to first nontrivial order S 1 1 cot 1 1 cos1q /1 sin/1 i 1q 1 sin1q /1. sin/1 8 We now show how to recover this expression from an analysis of the integral equations of the direct large- solution. Coupling an external source to the conduction electrons in the functional integral formulation of the model, it is easily seen that the conduction electron T matrix is given, in the large- limit, by T 1 Gi, 81 where G denotes the convolution of the fermion and auxiliary boson Green s function GG f G B. Hence, we have at T 8 a more sophisticated analysis. Equation 84 implies Im GA f A B, and hence Re S 1 1 A f A B / cosh (/). We make use of the expression C8 derived in Appendix C for the product of amplitudes A f A B and obtain Re S Re tan f i. After expressing in terms of q using Eq. 46 as tanh cot f tan 1q 1, Re S 1 coincides with the real part of Eq. 8. We now consider Im S 1, for which we need to go beyond the scaling limit and use global properties of the Green s functions. First expressing f as a convolution on the imaginary axis and using G (i) i () in the limit of a flat particle-hole symmetric band we obtain i GiAB 87 S 1 1 i Gi. 83 We first make use of the scaling limit of the two Green s functions, given by Eq. 9, and obtain the scaling form of G: with the definitions A dg f i f i, 88 A f A B GG f G B cosh /sin/. 84 We note that, in this scaling limit, the particle-hole asymmetry of the impurity Green s function has been lost altogether: has cancelled completely in the dependence of G in this limit and is only present in the prefactor. Thus, only Re S 1 can be extracted from the scaling limit, while Im S 1 requires B dg f ii f ig f i f i. 89 In the limit of vanishing, B() can be calculated from the scaling limit of G f. We obtain

13 386 PARCOLLET, GEORGES, KOTLIAR, AD SEGUPTA PRB 58 B 1 e i1q /1 e i f sin f i. 9 On the other hand, A contains high-frequency information that is lost in the scaling limit. We find A i 1 1q. 91 The details of these calculations are provided in Appendix E. Combining Eqs. 91, 9, 87, 83 we find agreement with the large limit of S 1 in Eq. 8. For a dilute array of impurities of concentration n imp, the conduction electronself energy is given by (i )n imp T(), to lowest order in n imp. As shown above, T() is given in the large- approach by the Fourier transform of G f ()G B (). The expansion 7 yields the long-time behavior G f () A f / fa () f / 4 f and G B ()A B / BA () B /. From the fact that f B 1, this implies G f ()G B ()1/1/ 1 f. Hence the resistivity behaves as 1Re S1 ct f, 9 where u is the impurity resistivity in the unitary limit. For the same reasons as above, the Anderson model result would lead to an exponent B in the regime 1. 7 Tn imp u D. The limit of a large number of channels We finally emphasize that all the expressions derived above greatly simplify in the limit of a large number of channels. This is expected, since in this limit the non- Fermi-liquid intermediate coupling fixed point becomes perturbatively accessible from the weak-coupling one. 1,1 The physics of the fixed point can be viewed as an almost free spin of size Qq weakly coupled to the conduction electrons. Indeed the large- expansion of the entropy 53, the Green function G f 33, and the twist 46 are S imp q ln q 1q ln1q q 1q, 6 e 1/ g f 1 cosh/ 1 ln sin, f 1 T T, ln q 1q, 93 and the leading terms in these expansions are given by the corresponding quantities for a free spin of size q. Moreover the scattering matrix S 1 and resistivity have the following expansion: Re S 1 1 q 1q, Im S 1 O 1 3, T/n imp u 1 q 1q, 94 while the anomalous dimensions read S f 1/ 1/, B 11/ 1/. VII. COCLUSIO In this paper, we have focused on the non-fermi-liquid overscreened regime of the SU()SU(K) multichannel Kondo model. This model has actually a wider range of possible behavior, which become apparent when other kinds of representations of the impurity spin are considered. In a recent short paper, 9 two of us have studied fully symmetric representations corresponding to Young tableaus with a single line of P boxes. This amounts to considering Schwinger bosons in place of the Abrikosov fermions used in the present work. It was demonstrated that, in that case, a transition occurs as a function of the size P of the impurity spin, from overscreening for PK to underscreening for PK, with an exactly screened point in between (PK). The large- analysis of the overscreened regime PK is essentially identical to that presented in the present paper for antisymmetric representations. Obviously, an interesting open problem is to understand the physics of the model for more general impurity spin representations, involving both bosonic and fermionic degrees of freedom corresponding respectively to the horizontal and vertical directions in the associated Young tableau. CFT methods are a precious guide in achieving this goal. In particular, the formulas and rules given in Appendixes A and B allow for an easy derivation of the impurity T residual entropy and zero-frequency T matrix, using Affleck and Ludwig s fusion principle and the identification of these quantities in terms of modular S matrices. An open question which certainly deserves further study is to identify which of these more general spin representations are such that a direct large- solution of the model can be found. This question has obvious potential applications to the multi-impurity problem and Kondo lattice models. During the course of this study, we learned of a work by A. Jerez,. Andrei and G. Zaránd on the same model using the Bethe Ansatz method. Our results and conclusions agree when a comparison is possible in particular for the impurity residual entropy and low-temperature behavior of physical quantities. ACKOWLEDGMETS We are most grateful to. Andrei for numerous discussions on the connections between the various approaches. We also acknowledge discussions with S. Sachdev at an early stage of this work about the importance of the thermal scaling functions. This work has been partly supported by a CRS-SF collaborative research Grant o. SF-IT- 1473COOP. Laboratoire de Physique Théorique de l Ecole ormale Supérieure is unité propre du CRS UP 71 associée à l ES et à l Université Paris-Sud. Work at Rutgers was also supported by SF Grant o